Problem 45 - Precalculus

Question

For the following exercises, use a graphing utility to find numerical or graphical evidence to determine the left and righthand limits of the function given as \(x\) approaches \(a\). If the function has a limit as \(x\) approaches \(a\), state it. If not, discuss why there is no limit. $$\lim _{x \rightarrow-1} \frac{|x+1|}{x+1}$$

Step-by-Step Solution

Answer

The limit does not exist because the left-hand and right-hand limits differ: \(-1\) from the left and \(1\) from the right.

Step 1: Understand the Expression

The function in question is \( \frac{|x+1|}{x+1} \). We need to evaluate the limits as \( x \) approaches \( -1 \). A graphing utility can help visualize the approach from both sides, but we can also analyze directly by considering the definition of absolute value.

Step 2: Analyze the Left-Hand Limit

For \( x \rightarrow -1^- \), \( x+1 \) is negative. Thus, \( |x+1| = -(x+1) \). The expression simplifies to \( \frac{-(x+1)}{x+1} = -1 \). So, \( \lim_{x \rightarrow -1^-} \frac{|x+1|}{x+1} = -1 \).

Step 3: Analyze the Right-Hand Limit

For \( x \rightarrow -1^+ \), \( x+1 \) is positive. Thus, \( |x+1| = x+1 \). The expression simplifies to \( \frac{x+1}{x+1} = 1 \). Hence, \( \lim_{x \rightarrow -1^+} \frac{|x+1|}{x+1} = 1 \).

Step 4: Conclusions on the Limit

The left-hand limit is \(-1\), while the right-hand limit is \(1\). Since these two are not the same, the two-sided limit \( \lim_{x \rightarrow -1} \frac{|x+1|}{x+1} \) does not exist. There is a discontinuity at \( x = -1 \) because the function behaves differently from the left and the right.

Key Concepts

Understanding Graphing Utility

A graphing utility is a tool, often a calculator or software, that allows us to visualize mathematical functions and their behaviors. It is incredibly useful for determining both numerical and graphical evidence, especially in situations where we are interested in finding limits of functions.
By plotting the function, you can observe how the function behaves as it approaches a specific value, such as when we approach \(x = -1\) in our exercise. This visualization underscores how the function behaves from both the left and the right side.

• Provides a clear representation of the function's behavior.
• Allows quick identification of discontinuity, such as jumps or holes.
• Aids in confirming computed limits with visual evidence.
• Helps in understanding more complex features of functions.

Graphing utilities are valuable not only for limits but also for understanding overall function behavior.

Exploring Left-Hand Limit

The left-hand limit describes how a function behaves as it approaches a particular value from the left side. It's an important concept in understanding the behavior of a function near points of interest.
For our specific function, as we approach \(x = -1^-\), we need to look at \(x\) values less than \(-1\). Here, the expression \(x + 1\) becomes negative, meaning we need to apply the absolute value as \(|x+1| = -(x+1)\).
Thus, simplifying gives us:

• \( \frac{-(x+1)}{x+1} = -1\).
• This results in the left-hand limit being \(-1\).

The left-hand limit helps us know one side of how the function behaves, demonstrating consistency in how we apply limits.

Understanding Right-Hand Limit

The right-hand limit gives insight into the behavior of a function as it approaches a specified value from the right side. This concept complements the left-hand limit to provide a complete picture of the function's behavior near the point of interest.
In our function, for \(x \rightarrow -1^+\), we consider \(x \) values greater than \(-1\). In this scenario, the expression \(x + 1\) is positive, directly equating to \(|x+1| = x+1\).
We simplify this to:

• \( \frac{x+1}{x+1} = 1\).
• The right-hand limit is therefore \(1\).

By evaluating the behavior from the right, we complement the left-hand approach, revealing the distinction at \(x = -1\), where the function displays discontinuity.

The Role of Absolute Value Function

The absolute value function is a fundamental mathematical concept represented by \(|x|\), which denotes the non-negative value of any real number \(x\). Its role is crucial in our exercise as it alters the behavior of the function \(\frac{|x+1|}{x+1}\) based on \(x+1\)'s sign.
This function serves two cases:

• When \(x+1\) is positive, \(|x+1| = x+1\).
• When \(x+1\) is negative, \(|x+1| = -(x+1)\).

In this exercise, understanding these cases enables us to comprehensively evaluate how the function behaves around the \(x = -1\) point.
It effectively influences how we calculate both the left-hand and right-hand limits, resulting in the conclusion that there is a discontinuity because the function does not have a consistent limit from both directions.